# Klein four-group geometry

I checked. Only the divisors of 24 produce the specific symmetry such that:

s=(n-k^2)/(2k)
s+k =(n+k^2)/(2k)
(s+k)^2 - s^2 = n
Jeremy, would you mind providing definitions for n, s and k?
n is clearly the automorphisms of the divisors of 24, but beyond that I am trying to decode in a precise way how you are getting your s's and k's.

Btw, in Lucas Sequence Notation where P = (s + k) and Q = (s * k) and D = P^2 - 4Q, the Discriminant, those 3 statements above can be reformulated, in order, via simple algebraic manipulation...

Q = (s * k) = ((n - k)^2)/(2)
P = (s + k) = ((n + k)^2)/(2k)
Q = (s * k) = ((n - k)^2)/(2)

sqrt (P^2 - 4Q), by the maths of Lucas Sequences, it follows, will always be an integer.

Also, could you clarify the below "claim" (as mathematicians call it) that "the square root produces the Klein four group from (n-k^2)/(2k)" (the first quoted mathematical statement above). What's apparent to you is not apparent to me. Don't forget that I'm self-taught too :-)
So one of the 4 Quantum "Spatial and angular momentum numbers" Ms where spin S = n-0.5
relative intensities = k(n+1-k)
http://en.wikipedia.org/wiki/Quantum_number#Spatial_and_angular_momentum_numbers

"the spin-raising operator is the square-root"

the square root produces the Klein four group from (n-k^2)/(2k)
- AC

P.S. As a little piece of mathematical trivia: D(P) = Q = 35 where P = (5 + 7) and Q = (5 * 7). What's the the next integer for which that relationship holds? Is it a common occurrence? Or an uncommon one?

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Feel free to add to this list Jeremy, ideally in your next post, so we can start to build a little resource to help you better communicate your ideas to professionals. No need for a thank you (You're welcome in advance.). - AC
Still AC, thank you very much for starting this. Mother Nature wreaked havoc in Ohio over the weekend, so I'll be busy for a while. I'll respond when I can. Thanks again.

Still AC, thank you very much for starting this. Mother Nature wreaked havoc in Ohio over the weekend, so I'll be busy for a while. I'll respond when I can. Thanks again.
Mapping freakish Mother Nature events to the prime number distribution. It's one potential application of your work, Jeremy. Just some food for thought. Hope all's well.

- AC

EDIT:
"for a set of complex numbers Z

Z=(((n+1)/2)-k) + i*sqrt(k(n+1-k)) whose (x/y) coordinates fall on a circle(x/y) or sphere(x/y/z) of radius (n+1)/2

so essentially those are points of a circle or sphere under quantization, integer and half-integer points.

a sphere would fall in line with the "inverse square law" perfectly for Spin, Energy and Force relative intensity coordinates.

make sense?"

Interesting view:
https://dl.dropbox.com/u/13155084/Recusion.png [Broken]

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EDIT:
"for a set of complex numbers Z

Z=(((n+1)/2)-k) + i*sqrt(k(n+1-k)) whose (x/y) coordinates fall on a circle(x/y) or sphere(x/y/z) of radius (n+1)/2

so essentially those are points of a circle or sphere under quantization, integer and half-integer points.

a sphere would fall in line with the "inverse square law" perfectly for Spin, Energy and Force relative intensity coordinates.

make sense?"
Related:
http://rxiv.org/pdf/1204.0102v1.pdf

Three (Related) Progressions of interest...

n^2 is Centered Hexagonal
http://oeis.org/A001570
Numerators of convergents to sqrt(3)/2
http://oeis.org/A144535
Denominators of continued fraction convergents to sqrt(12)
http://oeis.org/A041017

The 2nd progression is related to spin at the limit as n approaches infinity. The 3rd progression is the 1st difference between numerators and denominators. The 1 st progression is just every 2nd element of the 2nd.

- AC

This thread seems to have degenerated in a conversation between two people, while nobody else seems to know what is going on. I guess this thread can be safely locked. The conversation can probably be continued by PM if necessary.